Quantum Smeared Effects and the Stability of Matter

The intriguing role the smeared potential in special and the smearing effect in general play in optimization of the total energy and partition function of a quantum system opens the possibility analyzing the smearing phenomenon of the quantum fluctuation in a more fundamental way (Putz, 2009). 

In order to check Eq. (2.85) one separately computes eaeh of its side separately by the aid of -form of (2.74) and gets sueeessively the smearing average for wave-function  

Now for closely compare the expressions (2.86) and (2.87) the most elegant way is to make once more recourse to the smearing procedure, this time referring both to the entire paths and Feynman centroid to this end, the previous result (2.76) is here used explicitly as  

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

Based on the feet that the first two terms of Eq. (2.89) are equal due to the symmetry of the smearing average formula (2.74) at the interchange x while the mixed term of Eq. (2.88) expansion vanishes, = 0, in 

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